Rōpinepine Taparau

Hei Whakarāpopoto

I tēnei kōwae ako ka tūhurangia te āhua o te rōpinepine i ētahi taparau. He mea pūtake mai tēnei kōwae ako i te kōwae ako reo Ingarihi Fitness.

Achievement Objectives
2. Ka tautohu, ka whakamārama i ngā tikanga āhuahanga o ngā rōpū āhua ahu-2 me ngā rōpū āhua ahu-3.
7. Ka tautohu, ka whakamārama, ka whakamahi i ngā āhuatanga pūmau o ngā panoni.
Te Hononga ki te Marautanga

Taumata 4
Te Ine me te Āhuahanga

Whāinga Paetae 2 (Te Hanga):
Ka tautohu, ka whakamārama i ngā tikanga āhuahanga o ngā rōpū āhua ahu-2 me ngā rōpū āhua ahu-3.

Whāinga Paetae 7 (Te Panoni):
Ka tautohu, ka whakamārama, ka whakamahi i ngā āhuatanga pūmau o ngā panoni.

Ngā Whāinga Ako

Kia mōhio te ākonga ki:

  • te hanga, me te whakamārama i ngā rōpine-rite-takitahi me ngā rōpine-rite-takimaha
Ngā Tikanga Pāngarau
  • Kitea ai ngā momo rōpinepine i ngā wāhi maha. Ko ngā papariki o te papa kīhini, o te papa rūma horoi o ētahi whare ētahi wāhi e rua.
  • Ko te 360º te tapeke o ngā koki katoa huri noa i tētahi pū.
  • Hei tauira o te rōpine-rite-takimaha, ko ētahi tapawhā rite e rua me ētahi tapatoru rite e toru. Ko te 90º ngā koki roto o te tapawhā rite. Ko te 60º ngā koki roto o te tapatoru rite. Hui katoa, ko te 360º te tapeke: 90º + 90º + 60º + 60º + 60º = 360º
Ngā Rauemi
He ine-koki

He rūri

He huinga poro āhuatanga ake

Whārangi Tārua 1

Whārangi Tārua 2

He Rārangi Kupu
āhua ahu-2 2-dimensional shape
āhua ahu-3 3-dimensional shape
āhuatanga pūmau invariant property
āputa gap
ine-koki protractor
koki huripū whole angle, full turn (360º)
koki rārangi straight line angle (180º)
koki roto interior angle
panoni transformation
papariki tile
poro āhuatanga ake attribute blocks/shapes
putu degree
rōpinepine tessellation
rōpine-rite-takimaha semi-regular tessellation
rōpine-rite-takitahi regular tessellation
taparau rite regular polygon
tauwehe factor
Hei Raupapa I Ngā Mahi Ako
  1. Ko te mahi tuatahi, he tūhura i ngā koki roto o ngā momo taparau.

  2. Ngā tohutohu He tauira whakawhitinga kōrero
    Aratakina ngā ākonga kia whakarārangi i ngā koki roto o tētahi tapatoru rite, ka whakawhitiwhiti kōrero mō te tapeke o ngā koki roto.

    He aha tēnei momo tapatoru? He tapatoru rite nā te mea he ōrite te roa o ia tapa, he ōrite hoki ia kokonga.

    Titiro ki taku mahi. He tīhaehae i ia kokonga, me te whakatakoto kia piritahi. He aha tētahi mea e kitea ana i te takoto o ngā kokonga nei? He rārangi torotika ka hua mai.

    E hia putu i te rārangi torotika? Kotahi rau waru tekau putu.

    Nō reira e hia te tapeke o ngā koki roto o te tapatoru rite? Ko te 180 º

    Tukuna ngā ākonga kia tūhura mēnā ko te 180º te tapeke o ngā koki roto o ngā tapatoru katoa, ahakoa te āhua o aua tapatoru.

    Mā koutou e tuhi ētahi tapatoru, kia rerekē te āhua o tēnā me tēnā. Tapahia, ka tīhaehae i ngā kokonga. Whakatakotoria ngā kokonga o ia tapatoru kia kitea ai mēnā he rārangi torotika ka hua mai.

    He aha te whakakitenga? Ahakoa te āhua o te tapatoru, he rārangi torotika ka hua mai. Nō reira ko te 180º te tapeke o ngā koki roto o ngā momo tapatoru katoa.

    Tonoa ngā ākonga kia tūhura i te tapeke o ngā koki roto o ngā momo tapawhā.

    Ko tā koutou mahi ināianei, he tūhura i te tapeke o ngā koki roto o ngā momo tapawhā rerekē.

    Mā koutou e tuhi ētahi tapawhā, kia rerekē te āhua o tēnā me tēnā. Tapahia, ka tīhaehae i ngā kokonga. Whakatakotoria kia piritahi ngā kokonga o ia tapawhā. He aha te whakakitenga? Ahakoa te āhua o te tapawhā, he koki huripū ka hua mai. Nō reira ko te 360º te tapeke o ngā koki roto o ngā momo tapawhā katoa.

    Aratakina ngā ākonga kia tūhura i te tapeke o ngā koki o tētahi taparima.

    Titiro ki tēnei āhua. He tapawhā. E hia te tapeke o ōna koki roto? Ko te 360º.

    Mēnā ka tāpirihia tētahi tapatoru ki tētahi o ngā tapa, he aha te āhua ka hua mai? He taparima.

    Nō reira, ki ō whakaaro, e hia putu te tapeke o ngā koki roto o tētahi taparima? E 360º te tapawhā, kātahi ka tāpiria te 180º o te tapatoru, nō reira ko te 540º pea.

    Me pēhea te whakamātau i tēnei whakaaro. Mā te tuhi i tētahi taparima, me te whakamahi ine-koki hei ine i ngā koki roto katoa, kātahi ka tāpiria.

    Kia pērā anō te arataki i ngā ākonga kia tūhura i te tapeke o ngā koki roto o ētahi atu taparau. Ka whakaaturia ki tētahi tūtohi.

    Mēnā ka tāpiria tētahi anō tapatoru ki tētahi tapa o te taparima, he aha te āhua ka hua mai? He tapaono.

    Nō reira e hia te tapeke o ngā koki roto o tētahi tapaono? Ko te 540º o te taparima, tāpiria te 180º o te tapatoru, ko te 720º tērā.

    Tuhia te tapeke o ngā koki roto o ngā taparau ki tētahi tūtohi

    te taparau te tapeke o ōna koki roto
    tapatoru 180º
    tapawhā 360º
    taparima 540º
    tapaono 720º
    tapawhitu 900º
    ...  

    He aha te tauira e kitea ana i te tūtohi? Mo ia pikinga i te maha o ngā tapa, ka tāpiria te 180º.

    Tonoa ngā ākonga kia tātai i te rahi o ia koki roto o tēnā momo taparau rite o tēnā momo taparau rite.

    Mēnā he tapatoru rite te tapatoru, me pēhea te tātai i te rahi o ia koki roto? 180º te tapeke o ngā koki, ā, he ōrite te rahi o ia koki. Nō reira me whakawehe te 180º ki te 3, ka 60º.

    Tātaihia te rahi o ngā koki roto o ia momo taparau rite, ka tuhi atu ai ki te tūtohi. 

    te taparau te tapeke o ōna koki roto te rahi o ia koki o te taparau rite
    tapatoru 180º 180 ÷ 3 = 60º
    tapawhā 360º 360 ÷ 4 = 90º
    taparima 540º 540 ÷ 5 = 108º
    tapaono 720º 720 ÷ 6 = 120º
    tapawhitu 900º 900 ÷ 7 = 128.6º
    ...    
    Tonoa ngā ākonga kia tuhi i ētahi o ngā taparau rite.

    Ko tā koutou mahi, he tuhi i ētahi o ngā taparau rite. He aha ngā taputapu hei mahi māu? He ine-koki me he rūri.

    Me tīmata ki tētahi tapa, ka inea ngā koki ki ia pito o taua tapa. Hei tauira (mō tētahi wāhanga o te tapawaru rite):


  3. Ko te mahi tuarua, he tūhura i te rōpine-rite-takitahi o ngā momo taparau rite.

  4. Hoatu he huinga poro āhuatanga ake ki ia takirua ākonga.

    Whakamāramatia te tikanga o tēnei mea te rōpinepine.

    Tonoa ngā ākonga kia tūhura i ngā momo poro, kia kitea ai mēnā ka rōpinepine, kāore rānei e rōpinepine.

    Ko tā tātou mahi, he tūhura i tēnei mea te rōpinepine.

    Ākene pea he kupu hou te rōpinepine. Ko tōna tikanga mai rā anō, ko te whakanoho i ētahi mea kia piritata.

    Ko tōna tikanga pāngarau, ko te whakanoho i ētahi āhua (pēnei i ngā poro nei) kia piri, engari kaua tētahi e noho ki runga i tētahi, kia kaua hoki e kitea he āputa ki waenganui.

    Ko te mea māmā, ko te tapawhā rite. He rōpinepine ka puta i te whakanoho i ētahi tapawhā rite kia piri tahi:

    Tūhurangia ngā poro katoa. He aha ngā poro e hua mai ai he rōpinepine mēnā ka whakanoho kia piritahi? Tuhia he pikitia o ia rōpinepine ki te pepa inerite.

    Hei tauira:

    • te tapatoru rite:

    • te tapatoru waerite:

    • te taparara waerite:

    Whakaahuangia te Whārangi Tārua 1, ka tapahia ai i ngā taparau rite. Hoatu ki ia takirua ākonga.

    Tonoa rātou kia tūhura i te rōpinepine o ngā taparau rite:

    • He aha ngā taparau rite ka rōpinepine?
    • He aha ngā mea kāore e rōpinepine?
    • He aha pea te take e rōpinepine ai ētahi taparau rite, kāore ētahi?

    Anei ētahi taparau rite. Ka mahi takirua koutou ki te tūhura i te rōpinepine o ēnei taparau rite.

    Mēnā ka rōpinepine tētahi taparau rite, ka kīia tērā he rōpine-rite-takitahi. He aha ngā taparau rite e hua mai ai he rōpine-rite-takitahi? Ko te tapatoru rite, te tapawhā rite me te tapaono rite.

    Ka tapaina ēnei rōpinepine:

    • 3.3.3.3.3.3
    • 4.4.4.4
    • 6.6.6

    He aha ki ō whakaaro te tikanga o ēnei tapanga? Ko te 3.3.3.3.3.3 e tohu ana e ono ngā tapatoru rite huri noa i ia kokonga. Ko te 4.4.4.4 e tohu ana e whā ngā tapawhā rite huri noa i ia kokonga. Ko te 6.6.6 e tohu ana e toru ngā tapaono rite huri noa i ia kokonga.

    He aha ki ō whakaaro te take e rōpinepine ai te tapatoru rite, te tapawhā rite me te tapaono rite, kāore ētahi atu?

    E rua ngā take:

    1. Ko ngā koki roto o aua taparau rite, ka taea te whakawehe pū ki roto i te 360º. Arā:
      • Ngā koki roto o te tapatoru rite = 60º, 360 ÷ 60 = 60
      • Ngā koki roto o te tapawhā rite = 90º, 360 ÷ 90 = 4
      • Ngā koki roto o te tapaono rite = 120º, 360 ÷ 120 = 3
    2. He ōrite te roa o ngā tapa o ēnei taparau rite katoa.

  5. Ko te mahi tuatoru, he tūhura i te rōpine-rite-takimaha o ngā momo taparau rite.

 

Ngā tohutohu He tauira whakawhitinga kōrero
Aratakina ngā ākonga kia tūhura i te rōpine-rite-takimaha.

Arā anō ētahi rōpinepine ka puta i te whakamahinga o ētahi taparau rite e rua. Titiro ki te tūtohi e whakaatu ana i ngā koki roto o ngā momo taparau rite. Ki tō whakaaro, he aha ētahi taparau rite e rua e hua mai ai he rōpinepine? Kia rua ngā tapawhā rite, kia toru ngā tapatoru rite. Hui katoa ngā koki roto, ka 360º. Arā, 90 + 90 + 60 + 60 + 60 = 360.

Tuhia he pikitia o te rōpinepine o ētahi tapawhā rite e rua me ētahi tapatoru rite e toru. Me pēhea te whakanoho kia piritahi kia hua mai ai he rōpinepine?

Koia nei tētahi. Ko te 4.3.3.4.3 hei tapanga poto mō tēnei rōpinepine, nā te mea koia te raupapa o te whakanoho i ia kokonga – huri whakatekaraka i ia pū, he tapawhā rite, he tapatoru rite, he tapatoru rite, he tapawhā rite, he tapatoru rite.

Koia anō tētahi. Ko te 4.4.3.3.3 hei tapanga poto mō tēnei rōpinepine, nā te mea koia te raupapa o te whakanoho i ia kokonga – huri whakatekaraka i ia pū, he tapawhā rite, he tapawhā rite, he tapatoru rite, he tapatoru rite, he tapatoru rite.

Hoatu ngā taparau rite (Whārangi Tārua 1), ka tono ai i ngā ākonga kia tūhura i ngā rōpine-rite-takimaha.

Koia nei ngā rōpinepine e taea ana (Kei te Whārangi Tārua 2 ēnei rōpinepine e whakaaturia ana):

  • 3.3.3.4.4
  • 3.6.3.6
  • 3.3.4.3.4
  • 3.3.3.3.6
  • 4.8.8
  • 3.4.6.4
  • 3.12.12
  • 4.6.12

 

Anei ētahi taparau rite hei tūhura māu i ngā rōpinepine katoa e taea ana. Tuhia ngā rōpinepine ki te pepa, me te tapanga poto ki raro iho.

 

Hei Whakawhānui
Tukuna ngā ākonga kia tūhura i ētahi atu āhua e hua mai ai he rōpinepine. Hei tauira ēnei:

Kei ngā pae tukutuku nei ētahi atu tauira:
http://www.tessellations.org/eschergallery1thumbs.htm
http://www.google.co.nz/images?hl=en&q=examples+of+tessellations&um=1&ie=UTF-8&source=univ&ei=gGHsS8PbH5Hg7APe4pGPBg&sa=X&oi=image_result_group&ct=title&resnum=4&ved=0CDcQsAQwAw


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